(K, A g / N 0 3 , S04) Liquidus and
1699
Comparison with Theories
transition from B1 to I32 in the present study is in good l agreement with measurements made by Jamiesonlo Jand by Bassett.15 D. FezSi04-MgzSi04 System. Besides the alkali halides the only other solid solution series studied in detail as a function of pressure is the FezSi04-MgzSi04 system. Except for compositions of more than 80% MgzSiO4, members of this series undergo a phase transition from the olivine structure to the spinel structure. Experimental data of Akimoto and Fujisawal8 showed the separation of the subsolidus curves (Figure 6). The volume change, AVt, a t room temperature for pure FezSiO4 is approximately 4.4 cm3/mol and for MgzSiO4 it is estimated to be approximately 4 cm3/mol. At 800", the transition pressure, pt, for FezSiO4 is approximately 44 kbars and for MgzSiOd it is estimated to be approximately 127 kbars (Akimoto and Fujisawals). Using these data and ignoring the fact that olivine transforms to a different structure at greater than 80% MgZSiOl we calculated the subsolidus curves on the basis of eq 6 and 7. These are compared with the experimental data of Akimoto and Fujisawa in Figure 6. Considering the uncertainties in assumed A v t and pt, the agreement is remarkable.
Conclusions There is a striking paucity of experimental data for subsolidus relationships in P-x diagrams such as we have discussed in this paper. The fact that the equations we have proposed are in reasonable agreement with the data that are available leads us to believe that these equations should prove useful in interpreting subsolidus relationships when experimental data are lacking or insufficient. The RbC1-KC1 and KC1-NaC1 series are potentially useful for pressure calibration. It is possible to produce solid solutions in these systems which have a transition a t any desired pressure between 5 and 300 kbars simply by preparing the correct mole fraction. We have followed this procedure in our laboratory and found it to be successful when visual observations of the transition are possible.
Acknowledgments. This research was partially supported by NSF Grants GA-31902 and GA-38056X.
(18) S. Akimotoand H. Fujisawa,J. Geophys. Res., 73, 1467 (1968).
Thermodynamic Properties of the Reciprocal System (K+, Ag+l IN03-, S 0 4 2 - ) from Its Phase Diagram M. L. Saboungi,Ia C. Vallet,*lb and Y. Doucet Laboratoire de Thermodynamiqoe des Sels Fondos, associb au C. N. R.S., Centre St. Jbrbme, 13013 MarseiUe, France (Received May 15, 1972) Publication costs assisted by the University of Provence, Marseille. France
The liquidus curves calculated from two solution theories are compared with the values obtained from our new phase diagram data on the (K+,Ag+I IN03-,S042-) reciprocal salt system. Calculations from a random mixing model and a nonrandom mixing quasichemical model are given. In dilute solution of K2SO4 in AgN03, the pair formation energy A A 1 is determined by comparison between the expressions resulting from the generalized quasilattice theory and the quasichemical model. The value obtained is in accordance with the determination of Watt and Blander. In more concentrated solutions, the nonrandom mixing assumption leads to results which are close to our experimental data on both diagonals.
I. Introduction
11. Theories of Reciprocal Molten Salt Systems
The thermodynamic study of the molten salt reciprocal mixture (Ag+,K+, ' N o s - ,Sod2-) has been carried out from its liquidus curves, corresponding to both pairs AgN 0 ~ ( s o l v e n t ) - K ~ Sand 0 ~ KN03(solvent)-AgzS04. The departure from ideality, as expressed in terms of the mole fraction activity coefficients relative to pure solvent, is small, but not negligible. It is, however, greater for the solution of silver sulfate in potassium nitrate than for the solution of potassium sulfate in silver nitrate.2 This work is an attempt at calculating those deviations from ideality by taking into account short-range interactions.
In this section, we will give a short review of two principal theories: the Flood, Ferland, and Grjotheim theory3 (FFG) based on the hypothesis of a random mixing, and the quasichemical theory where a nonrandom mixing is considered. Both theories are extended to asymmetrical mixtures and put into a useful form for our calculations. (1) (a) Research Fellow C.N.R.S., Lebanon, (b) Reactor Chemistry Division, Oak Ridge Laboratory, Oak Ridge, Tn. 37830. (2) (a) C. Valiet and M. L. Saboungi, C. R. Acad. Sci., Ser. C, 272, 146 (1971); (b) M. L. Saboungi, These de Specialite Aix-Marseille, 1971. (3) H. Flood, T. Fplrland, and K. Grjotheim, Z.Anorg. AUg. Chem., 276, 289 (1954). The Journal of Physical Chemistry, Vol. 77, No. 73, 7973
M. L. Saboungi, C. Vallet, and Y . Doucet
1700
Random Mixing. The Flood, Fqirland, and Grjotheim theory has been applied to reciprocal mixtures of the general type (W+,X+/IY-,Z2-) with the following basic assumptions. (a) The configurational entropy of the solution is supposed to be the same as in an ideal mixture. The cations are randomly distributed on cation sites and the anions on anion sitese4If the system contains w mol of W + , x mol of X + , y mol of Y-, and 2 mol of 22-, the configurational entropy is A S o = R ( w In ( w x)/w x In ( w x)/x +
+
+
+
Y In 0, + Z)/Y + z In 0, + z ) / z ) (1) (b) Long-range interactions are neglected. Only interactions between nearest and next nearest neighbors of both different charge sign and the same charge sign are taken into account. The former are due to the W f Y - , X f Y - , W’Z0.5 , X+Zo.s- contacts. The two last types of contact are such that the pairs in question are neutral. In order to find the total number of contacts, the method used by Guggenheim5 for mixtures of neutral molecules has been followed, so that the relative numbers corresponding the various contacts are respectively
+
x)
+ +
+
+
xy
2xwzl(w
+
x)2
2yxz/(w
+
x y
+
xu. +-
2xyz (w X
+
+
( w + x ) 3 Xz (4)
awY = fwvXwX. (5) where Xw and Xy are the Temkin ionic fractions and UWY is the activity of WY.
xw
w/(w
=
+
x, = Y/b + x, = x / ( w + x,
=
x) 2)
(6)
x)
+ 2) + x = y + 22. From eq 4
210,
hZ(X,’
+
- X,’)] + (XWXZ’ + xyxx’)(xw’xz’
+ XX‘X,’) (7)
The Xi’ being the equivalent ionic fractions
XX’ =
x/(w +x)
Xk.’
=
y/(y+22)
X,.‘
=
w/(w
+XI
+
X,’ = 2 z / (Y 2 2 ) and AGO the free energy change for the exchange reaction
The interaction parameter which has been experimentally determined previously in the case of common ion systems6 may be interpreted in terms of the variation of energy corresponding to the formation of the above associations. I t can be defined as the partial molal enthalpy of mixing of the solute a t infinite dilution; in the case of a regular mixture, it is also the interaction parameter found from the Gibbs free energy of mixing. The Gibbs free energy of mixing can be written as follows
2wyz (w XI2
(w+x)3
By definition the activity coefficient f w y is
xy
2wzy/(w
+XI3
+
+
x)
2xz/(w x) To represent the enthalpy of the solution, one must include interactions such as W+-Y--X+; Y--W+-Z0.5-; W+-Z0,5--X+; Y--X+-Zo,5-. The corresponding relative numbers of bonds may be calculated in the same manner. wxy/(w
(w x)3
the neutrality condition being w and 5 , the excess free energy of WY in the WY-XZZ mixture is T In f W Y = Xx’Xz’ [AGO XX(Xz’- X s ’ )
WYl(W + x ) XY/(W 2wz/(w
++ 2 w i ) xw 2 x( 4w2 2 - Y ) Ax + 2 x z ( x - w) x(xy + 2wz)
24X.y
Y
AX - TAS” ( 2 )
where AS‘ is given by eq 1, the Gij“ terms are relative to the pure components ij and the Xi’s represent the interaction constants for the binary systems with a common ion, the single subscript indicating the common ion. The chemical potential of a component, WY, for instance is
W Y (liq)
+
‘12
X ~ Z (liq)
X Y (liq)
+
‘ i z W,Z (lis)
(9)
For the cases of interest here w = y and x = 22. Consequently we may write in terms of Xx’
RT In f w y
= X,’2(AGo
+ 2Xw + 2X, - Ax - Xz) + 2xx’3(xx + xz - xw - A,) (10)
Nonrandom Mixing. In a second approach, the dissimiliarities between the properties of the ions have been taken into account; a nonrandom mixing has therefore been assumed. If y* is the departure from random mixing, the number of X-Y bonds is [ x y / ( w + x ) ~ ]+ y*, by counting only nearest neighbors pairs. The energy change for the reaction
+
X-Z,,
pair
X-Ypair
+
W-ZOSpair (11) is denoted by A F / Z , Z being the mean coordination number of the ions in the molten salt. Applying the massW-Ypalr
(4) M. Temkin, Acta. Phys. Chim. USSR. 20, 41 1 (1945)
(5) E. A . Guggenheim. “Mixtures,” Oxford University Press, London, 1952. (6) C. Vaiiet, These Aix-Marseille, 1970.
The Journal of Physical Chemistry, Vol. 77, No. 13, 1973
1701
(K, Ag/N03, SO4) Liquidus and Comparison with Theories
action law to the above reaction as in the quasichemical theory we obtain
2wz
= K =
WY
- Y*]
Using the equivalent ionic fractions given by eq 6, we may write
..*
r
r
1
If y* is smaller than any of the products of an anionic and cationic equivalent fraction, the logarithmic terms may be expanded in powers of y*. Hence neglecting the square terms y * = - X w ' X x ' X y ' X z ' (AE'IZRT) (14)' The excess energy of mixing due to the nonrandom distribution is A E * = y*AE' = - X,'X,'XZ'X,'[(A EO)'/ZRT] (15)
and the corresponding free excess energy of mixing is AG* = T
L1
AE*d(l/T) =
-
Xx'Xw'Xz'XY' [(AE0)2/2ZRT] (16) The total free energy of the solution, which takes into account the departure from a random distribution, is
+
+ +
G*'- = G M (w x)AG* (17) where GM is given by eq 2 and ( w x) is the number of moles present in the solution. By following the same derivations as in eq 3 and 5 we obtain
+ 2Xw + 2Xy - Ax + 2Xxf3(Xx+ - Xw - + 2A) - 3 X l 4 A
R T In f w Y = X,'2(AG"
Xz - A )
XZ
Xy
(18) where A is given by
-1= - (AEo)'/2ZRT
-
-
(AG0)'/22RT
Merck reagent grade chemicals were used. The salts were finely ground (except in the case of AgN03) and heat dried. The relative error on the determination of solute masses did not exceed Each run was begun by a measurement of the melting point of the pure solvent. The solute was then added in three or four successive portions in each crucible and the liquidus temperature determined from the thermal halt. The quantities added were increasingly significant in determining the composition, so that the final composition was known with the same relative precision. The temperature was recorded with a Pt-Pt-10% Rh thermocouple. The hot junction, sealed in a thin-walled Pyrex tube, was introduced along the axis of the cylindrical crucible and the cold junction was in a water triple point cell. The composition and temperature in the sample were kept homogeneous by efficient stirring. The major fraction of the thermocouple emf was compensated by means of a P 12 manual potentiometer and the remainder, less than 1 mV, was amplified with a Leeds and Northrup voltage amplifier and recorded on a MECI potentiometer. The cooling rate being less than 1 K min-1, the cooling solid were curves recorded for the liquid and the liquid nearly two straight lines of different slopes. The temperature at which solid forms is given by their i n t e r s e ~ t i o n . ~ Each recording was performed in triplicate and the reproducibility was better than 0.02 K.
+
IV. Results The activity coefficients of the solvent on both diagonals were calculated, under the assumption of no formation of solid solution from the liquidus temperatures of the phase diagram2b utilizing the Schroder van Laar equation. : R l n a = -AH,
111. Experimental Section In order to study the reciprocal diagram of the system (K+,Ag+lIN03-,S042-), we determined the liquidus curves on both (AgN03-K&04) and (KN03-Ag2S04) diagonals.
T
(20) AH0 is the heat of fusion of the solvent a t the melting temperature TOand ACp, the heat capacity change upon melting, is defined by AC, = C,(liq) - C,(sol) The parameters used in eq 20 were obtained from standard literature values.10-11 Activity coefficients were calculated for each solvent from the expressions
(19)
A different statistical mechanical treatment of the mixture ( W + , X + //Y-,Z2-) may be used by applying the conformal ionic solution (CIS) theory proposed by Blander and Yosim.7 Depending on the order of the approximation in the power expansion, one can derive easily either eq 10 (first-order approximation) or 18 (second-order approximation). The nonrandom mixing term A, however, cannot be obtained readily from theory but is proportional to It is introduced by comparison with the corresponding term in the quasilattice theory.8
(,+, --?i,) + AC, (3-1 - In
aKNO,
=
XKXN03fKN03
and
(21) aAgN03
= XAgXN0,fAgNo3
where the Xi's are the ionic fractions defined in the eq 6. The values of R T In fKNO3 and R T In fAgNO3 are plotted in Figures 1and 2, respectively. Calculations in Concentrated Solutions. We will make two sets of calculation from theory for both diagonals for comparison with the experimental data. The first calculation. is based upon the FFG theory and consists of applying eq 10. The second calculation is based upon the (7) M . Blander and S. J. Yosim, J. Chem. Phys., 39,2610 (1963). (8) M. Blander, J. Chem. Phys., 34, 432 (1961);M. Blander and J. Braunstein, Ann. N. Y . Acad. Sci., 79,838 (1960). (9) C.Vallet, J. Chim. Phys. F, 69,311 (1972). (IO) K. K. Kelley, Bureau of Mines, Bulletin No. 584,393,477. (11) F. D. Rossini, Nat. Bur. Stand. U . S. Circ., 500 (1952). The Journal of Physical Chemistry, Vol. 77, No. 13, 1973
1702
M. L. Saboungi, C.
Vallet, and Y . Doucet
0
-5
n
0
D
2
-
-10
l-
-45
-20 5
0
10 X
Figure 1. Experimental and calculated values at the liquidus temperatures of excess chemical potential of K N 0 3 vs. X A ~ ’ * in the KN03-Ag2S04 quasibinary mixture: (circle) our experimental results; FFG, Flood, Fglriand, and Grjotheim theory (random mixing); CIS, conformal ionic solution theory (nonrandom mixing). Inset shows our results in the dilute solution range and the corresponding limiting slopes.
Figure 2. Experimental and calculated values at the liquidus temperatures of excess chemical potential of A g N 0 3 vs. X K ’ ~in the AgN03-K2S04 quasibinary mixture: symbols, same as in Figure 1. TABLE I: Values of X i for Binary Mixtures
quasichemical theory or the CIS theory using eq 18. A comparison of these calculations with our experimental data will enable us to evaluate the importance of the nonrandom mixing term. The Gibbs free energy exchange, AGO, corresponding to the equilibrium K N O B (liq)
+
‘/zAg2S0, (liq) A g N 0 3 (liq)
‘/&SO4
(liq)
has been calculated from tabulated data on pure molten saltsll by taking into account the heats of fusion, formation, and transition. The specific heats of the reactants and the products are assumed’ t o be equal.12 The value obtained is 4 kcal/mol. The variation of AG” with temperature along the liquidus is negligible. We reported in Table I the values of the interaction parameters. These were estimated from phase diagrams or calorimetric data when the latter were available. By applying eq 20 to phase diagram data on the AgN03Ag2S04I3 and KN03-KzS0414 systems, average values for XAg and XK were then estimated by assuming a regular 3 was deterbehavior of the solutions. The X ~ 0 coefficient mined by comparison with the calorimetric data of Kleppa15 and the results of Franzosini.16 ‘The KzS04AgzS0417 system forms solid solutions; the As04 parameter is consequently small and might be assumed equal to zero.18 A mean value of the quasilattice “coordination number” Z has been taken equal to 5 as it varies generally from 4 to 6 in some molten ~ a 1 t s . l ~ As discussed above, values of AGO, A,, and Z were utilized in conjunction with eq 10, 18, and 19 to calculate the The Journai of Physical Chemistry, Vol. 77, No. 13, 1973
Mixture
Xi, cai/mol
Ag NO3-K N Osa AgN03-Ag2S0db KN03-K2S04C
-500 700
K2S04-Ag2S0dd
@
f
1;
*
From ref 15 and 16. From ref 13. From ref 14.
300 0 From ref 17.
two sets of values of RT In f s o l v along each diagonal. These values are reported in Tables I1 and 111. In Figure 1, experimental values of RT In fKNO3 have been plotted us. XAgf2and compared t o those calculated from both theories. At the dilute end, the CIS theory is in very good agreement with the experimental data. In fact, if the parameters were adjusted within their known uncertainties, an exact fit could be obtained along the curve. In the case of the AgN03-KzS04 quasibinary mixture, the three sets of values of RT In fAgNO3 have been plotted us. X K ’ ~A . striking agreement is obtained between our experimental values and those calculated from the CIS theory, while a large departure from the values calculated fromkhe FFG theory is evident. (12) M. Blander and L. E. Topoi, Inorg. Chem., 5, 1641 (1966). (13) E. Kordes, 2. Elektrochem., 55, 600 (1951). (14) E. Ph. Perman and W. J. Howells, J. Chem. SOC., 123, 2128 (1923). (15) 0. J. Kieppa, R . B. Clarke, and L. S. Hersh, J. Chem. Phys., 35, 175 (1961). (16) P. Franzosini and C. Sinistri, Ric. Sci., 439 (1963). (17) R . Nacken, Neues Jahrb. Mineral. Geol. Palaontol. Beil. Band., 24, 1 (1907). (18) M. Blanderand L. E. Topol, Electrochim. Acta, 10, 1161 (1965). (19) H. A. Levy, P. A. Agron, M. A. Bredig, and H. D. Danford Ann. N. Y. Acad. Sci., 79, 762 (1960).
(K, A g / N 0 3 , SO4) Liquidus and Comparison with Theories
TABLE 11: Excess Chemical Potentials of KN03, along the Liquidus, in t h e KN03-AgzS04 Quasibinary RT In ~ Io 4 x A q f 2
Measured
22 40 46 51 58 63 73 103 122 147 160 171 195 21 2 235 254
12.929 19.243 23.275 25.558 27.641 29.470 32.302 42.618 47.877 54.848 57.750 61.368 65.61 7 68.700 73.040 77.263
~
Nonrandom mixing (CIS)
6.470 12.050 13.880 15.420 17.630 19.170 22.250 31.760 37.810 45.870
11.770 16.920 18.758 20.626 23.336 25.104 29.012 40.300 47.330 56.370 61.01 0 64.360 75.590 79.480 87.370 93.070
50.000 53.640 61.410 67.060 74.630 80.950
~
I 04xK'2
Measured
Random mixing (FFG)
Nonrandom mixing (CIS)
20.5 24.5 29.6 37.9 45.7 53.6 65 83.9 94.1 106.2 125.6 144.7
-4.201 -4.657 -5.293 -6.470 - 7.462 -8.446 -9.595 - 1 1.798 - 12.932 - 14.232 - 15.943 -17.814
-8.180 -9.340 11.670 -14.770 - 17.880 - 20.980 -25.250 -32.61 0 -36.480 -41.120 -48.860 -56.200
-3.984 -4.496 -5.086 -6.466 -7.846 -8.870 10.205 -1 2.068. 13.470 -14.570 16.646 - 18.773
-
2.2
I
1
I
I
400
450
K N O ~
Random mixing (FFG)
TABLE Ill: Excess Chemical Potentials of AgN03, along the Liquidus, in the AgN03-KzS04 Quasibinary
~~
1703
-
Consequently, at intermediate concentrations, a nonrandom mixing hypothesis is necessary for describing solution behavior.
Calculations in Dilute Solutions. The AgS04- Pair Formation Energy Although phase diagram measurements are not among the most sensitive for the study of association of dilute solutes and although such a study was not the original aim of our investigation, we felt that it would be worthwhile to estimate the pair formation energy of AgS04 from our lowest concentration measurements. This seemed of particular interest because of a discrepancy in reported results for the temperature dependence of the AgS04- pair formation energy in KN03. The very dilute solution range has been investigated by Watt and Blander20 and by Sacchetto, et a1.,21 by means of emf measurement. These authors discussed the properties of very dilute solution of K2S04 and AgN03 in the solvent KNOJ in terms of the quasilattice model. According to the quasilattice calculations22.2a in both the symmetric and asymmetric approximations the conven-
."
I
350 I
(OC)
Figure 3. Extrapolation of the energy of ion pair formation AA, from results of Watt and Blander.*O
tional association constant K1 is given by
K,
= Z[exp(-AAJRT)
- 11
(22)
where 2 is the mean coordination number in the quasilattice and AA1 is a pair formation energy. If the yi are Temkin ionic fractions of species, considering AgS04- pair formation, the association constant is defined as Kl = YAgSO,/YAgYSO, (23) The part of the K N 0 3 activity coefficient arising from the nearest neighbor interactions which lead to nonrandom mixing can be written as By expanding In fKN03', one obtains easily the limiting equation
which may be applied to the data in very dilute solutions. In order to account for the influence of longer range interactions on the excess chemical potential of KN03, we have added the binary terms included in eq 10 as suggested by Blander and Topol.18 The association constant, K1, can then be calculated from the limiting slope [ls] of the plot of excess chemical potential of K N 0 3 plotted us. the square of the ionic mole fraction of silver, X A ~ ' [IS] =
(RT/2)K,
+
(2AK
+
2AN0, -
- /Zso4)
(26) From the results given in Table IV and represented in inset of Figure 1, the estimated value of the limiting slope is in the range 7300-5500. Our resultant value for the pair formation energy, AA1, at 607 K24 has an estimated uncertainty of 150 cal. It should be noted that this accuracy is better than might be expected for data obtained from phase diagram measurements at such low concentrations. (20) W. J. Watt and M. Blander, J. Phys. Chem., 64,729 (1960). (21) G. A. Sacchetto, C . Macca, and G. G. Bombi, J. ElectroanaL Chem. 36,319 (1972). (22) M. Blander, J. Phys. Chem., 63, 1262 (1959). (23) J. Braunstein, "Ionic Interactions," Vol. I, Academic Press, New York, N, Y., 1971, p232. (24) We notice that the AA1 variation along the liquidus branch (the freezing point depression being of the order of 5 K) is negligible. We assume consequently that our AA, determination has been obtained at the melting temperature of pure KN03 (607 K) The Journalof Physical Chemistry, Vol. 77, No. 13, 1973
Zao-Shon Liang and William I . Higuchi
1704
TABLE IV: Excess Chemical Potential of KNO3, along the Liquidus, in the Dilute End of KN03-AgzS04
1o4xAp12 RTln f K N O J
1 2 5 7 9 12 17 0.714 1.245 2.452 3.527 4.566 6.617 9.372
TABLE V: Comparison of Values of AgS04- Pair Formation Energy in Molten KNO3 ~
Z 4
- A A I , ~kcal - AA l , b kcal - AA 1 ,c kcal
1.51 F 0.15
6
5
1.35
0.15
1.6
1.19F 0.15 1.2
1.58 0.05
Values extrapolated at 607 K from ref 20 assuming a a This work linear temperature dependence. Average value from ref 21 in the temperature range 622-71 5 K assuming no temperature dependence.
Comparison of our results for 2 = 4, 5, and 6 with those previously reported,20,21 is given in Table V. Our value of A A I is in better agreement with a value extrapolated (Figure 3) from the reported results of Watt and Blan-
der20 than with the temperature independent value reported by Sacchetto, e t al.21 Nevertheless, a more specific investigation with a more sensitive experimental method is required to provide an unequivocal evaluation of the temperature dependence of the association energy.
Conclusion Experimental data have been presented for the system (K+,Ag+1 IN03-,S042-) and have been interpreted theoretically. In dilute solutions, our estimation of the (AgS04)- pair formation energy is in the same range of order as previous determinations. (The nonrandom mixing hypothesis was found to give better agreement with the experimental results than the random mixing hypothesis, the fit being most striking a t high concentrations.)
Acknowledgments. The authors wish to thank Dr. M. Blander (Argonne National Laboratory) and Dr. J. Braunstein (Oak Ridge National Laboratory) for their interest and for many helpful suggestions. Financial assistance from the National Research Council of Lebanon to one of us (M. L. S.) is gratefully acknowledged.
Kinetics and Mechanism of the Reaction between Hydroxyapatite and Fluoride in Aqueous Acidic Medial y2
Zao-Shon Liang and William I . Higuchi* College of Pharmacy, The University of Michigan, Ann Arbor, Michigan 48704 (Received March 24, 7972) Publication costs assisted by the National lnstitute of Dental Research
The physical model of the reaction was investigated both experimentally and theoretically over a wide range of conditions. The theoretical rates were calculated as a function of pH, fluoride concentration, and phosphate concentration of the reaction solution. Experimental rate data obtained under a wide variety of conditions were compared to these model predictions. The agreement of the model predictions with the experiments was found to be essentially quantitative under all conditions with the adjustment of a single parameter, the apatite-calcium fluoride “equilibrium” constant. The magnitude of this constant was found to be consistent with the interpretation that the apatite-calcium fluoride interface was significantly supersaturated with respect to calcium fluoride during the reaction. A calcium fluoride activity product of about 10-8 was deduced from this analysis. This is greater than the solubility product of calcium flouride by a factor of about 103. Supersaturations of this same order of magnitude were found in the calcium fluoride precipitation studies.
Introduction There is overwhelming evidence today that the topical Of concentrated fluoride to teeth results in the significant reduction in the tendency for dental caries formation.3 While certain aspects of the chemistry of the fluoride-enamel (the mineral of which is principally hydroxyapatite) reactions have been investiand gated, these studies have not considered the mechanisms of the reaction.4 The Journal of Physical Chemistry, Vol. 77, No. 13, 7973
When hydroxyapatite is exposed to high concentrations of fluoride, calcium fluoride is precipitated while the (1) Presented at 49th Meeting of international Association for Dental Research, Chicago, Ill., March 1971. (2) Portions of this paper are derived from the Ph.D. dissertation of 2.
Liang, The University of Michigan, 1971. (3) F, Brudevold, A. Savory, D. E. Gardner, M. Spinelii, and R. Speirs, Arch. Oral Biol., 8, 167 (1963); J. R. Mellberg, J. Dent. Res., 45, 303 (1966). (4) H. G. McCann, Arch. Ora/ Bioi., 13, 987 (1966); A. Maiaowalla and H. M. Myers, J. Dent. Res., 41, 413 (1962).
